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Complex Numbers

by Robert Waltz

Rated: 18+ · Book · Personal · #1196512

Not for the faint of art.

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

Elisa the Bunny Stik

about 6 hours ago
In response to "Pleasure and Pain"

it illustrates why it's important to have people from different backgrounds, with different worldviews, working on scientific research.

This right here is why I get rather irritated by people poo-pooing college these days. I feel like that attitude effectively shits on minority populations that may not have strong finances but do possess persistent curiosity and want to learn about various subjects in-depth. In the 90s, these groups were exclusively told to go to trade schools even if they had the aptitude for college education. Nowadays the blanket "Everyone should consider a trade" mantra employs a more subtle (but still abundant) form of discrimination. I'm so over it.

1 Like

Kåre Enga in Udon Thani

Apr 25, 2024 at 3:16am
In response to "Going to the Dogs"

I have ants in Thailand. Will I miss them in Montana? No.

Would it be nice to have a cat and a dog, here or there? Yes, but I have struggles keeping my neglected plants alive.

Not every situation is suitable for everything or everybody.

Kåre Enga in Udon Thani

Apr 25, 2024 at 3:09am
In response to "Tipped Out"

I don't tip. In my travels I rarely find tipping the norm.

I also don't haggle.

When you add the tip... the true cost of America becomes evident. But what would I know? I travel while others who have double my income can't pay rent.

Kåre Enga in Udon Thani

Apr 25, 2024 at 2:58am
In response to "Reality Cubed"

My dreams may be different from your dreams but if we share them with others then some form of the dream may live on. Regardless, if we don't eat we die. "Joy gives us wings." "Reality bites."

Aiva Raine

Apr 24, 2024 at 4:26pm
In response to "Tipped Out"

Agreed. I'll tip and well, for actual waitstaff at restaurants. I'll tip appropriately for bell-staff who assists with luggage at hotels, the more luggage, the bigger the tip. I'll give a tip to the valet as well. Beyond that- I don't tip. Certainly no tip for the fast food workers, coffee baristas, grocery store clerks, etc.

It annoys me to no end that we're being asked to tip for all kinds of jobs where they are being paid an hourly wage. I've worked as a waitress when I was young and my daughter currently does. In those jobs- she absolutely depends upon proper tipping to make a decent wage. And I hate the fact that she is dependent upon tips to do so.

It leaves women especially at the mercy of arsh**() on whether she wants to accept hostile, negative, sexist, harassing, demeaning, and negative behavior from her customers in order to make a living. It condones people treating their waitstaff like shite in order to get paid. While there is much improvement from the 30 years ago when I was working as a waitress in the skimpy outfits we were forced to wear- it's still an issue at non-chain, independent owned restaurants and diners.

Women are left to the mercies of jerk customers and their only recourse is the manager who they have to hope supports their waitstaff rather than the jerk customer. Tipping should be eliminated. All employees should be paid a livable wage and waitstaff especially should not be dependent upon tolerating crappy to obscene behavior from customers in order to make a living.

But, until that happens- I always tip at least 20% and often times more. Very rarely have I tipped lower and the service in those situations was abysmal and they still got 10%.

1 Like

GeminiGem of House Lannister

Apr 23, 2024 at 6:51pm
In response to "Going to the Dogs"

I don't understand the intersection, either. I lived in a much smaller city and found that having dogs was a hassle (although it didn't stop me), what with the complainey neighbors, the extra licensing, etc. I live in a rural location now and it works great. I can't wrap my head around living in NYC with or without a dog, although I like the idea of a place that issues a canine and supplies upon moving there. Can that please be a thing?

1 Like

Lazy Writer est 4/24/2008

Apr 23, 2024 at 3:31pm
In response to "Going to the Dogs"

I love my little dog...but I live in a biggish house with a fenced backyard.

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:14pm
In response to "That's Cole"

I knew and ate Kohlrabi before I ever heard of kale. Kohlrabi is so much better in every way.

That last quoted sentence is hilarious. I am glad that I wasn't bred to be a Brussels Sprout.

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:11pm
In response to "Cash Stash"

The discussion of money is an interesting one. Remember the recent collapse of a few banks? I think it was about a year or so ago?
Yes, deposits were FDIC insured up to $250,000 per depositor. But some people and companies had more money in that bank than the FDIC was insuring.
I never followed up on what happened to those people. I don't run in circles of people who own that much cash.

My accounting professor took half a period to explain to us what happened and how the bank went from Silicone Valley darling to dead so fast.

He also told us that the value we give money is not backed by anything except our hope and faith that our economy will keep going.

We didn't discuss whether the value of money needed a lower income class of people.

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:04pm
In response to "Revisited: "Hack This""

In my last psychology class, we discussed the evolution of the brain. It is now settled science that we evolved to have not only larger, but also vastly more powerful brains than most other animals because we learned how to cook (process) our food.
Although we are only about 1/4 to 1/3 of a normal gorilla, we eat half of the calories in a day. Not only that, we prepare the food in a way that we get to use more of the calories that we ingest.
The result is that our brain takes up more space in our noggin than a gorilla brain takes up in his noggin. Not only that, each of our brain cells is smaller, so we can fit exponentially more of those into our head.

Now, using all that goodness ...

1 Like

August 16, 2020 at 12:04am

#990910

Changes in Attitude

Ever changed your mind?

https://www.bbc.com/future/article/20180622-the-surprising-reason-people-change-...

The surprising reason people change their minds
We usually believe that our opinions are stubborn and fixed. But new research shows that our views, even on politics, are changing all the time – just not for the reasons you’d expect.

You'd never know it from perusing social media.

Wherever you look at the moment, we seem divided – Brexiteer or Remainer, pro-President Trump or against. And no matter how much we argue, none of us appear to change our minds.

Article is from 2018, so some of the examples may seem dated, since we seem to be down to two issues: Whether to wear a mask or be an idiot, and whether people should have equal treatment under the law or not.

No one's changing their minds about those issues, either.

But new research suggests that, in fact, we can let go of our opinions – and that opposition can even turn into acceptance.

However, I know that my views on some things have changed over the years, so from that single data point, I can extrapolate that this is probably true. It's not like I have special mind-changing powers that no one else does.

For decades, research on confirmation bias has shown that we are more likely to look out for, notice and remember anything that confirms opinions we already hold.

Like, for example, me and this article.

If you like drinking wine, you’re more likely to remember the occasional studies which find a benefit from alcohol than the research on its risks.

While I can accept that this is generally true, it's not that I'm not aware of the risks of drinking alcohol; it's just that, for me, the benefits outweigh the risks. Besides, I have yet to see a "risk" study that takes into account how delicious it is.

Our brains are also faster at processing opinions we agree with.

This is one of those things that probably didn't need a formal study, but I suppose it's good that they've delved into it.

All of this would suggest that we hold our opinions dear. This is true, but it doesn’t mean those opinions are fixed forever. We are more fickle than we think.

I know I have to catch myself when I start thinking, "I've always held this opinion," when it's clear that I haven't; I just don't want to admit that I was wrong, so I try to convince myself that I was always right. But for me, the only thing worse than being wrong is being pigheadedly stubborn like that.

In other words, we rationalise the things we feel stuck with. It’s as though we free up brain space to get on with our lives by deciding it’s not so bad, after all.

Yeah, I'm still not there with this whole pandemic thing.

Next, Laurin looked at views on Ontario’s 2015 ban on smoking in parks and restaurant patios. She found that people didn’t only change their opinions after the ban had been brought in – they changed what they remembered about their own behaviour.

This is like what I said above, and in previous entries here: we can't really trust our own memories sometimes.

So it’s not that people simply become accustomed to a new situation. Instead, they actually change their thinking. It is as though they can’t bear to continue feeling angry, so they subconsciously look for ways to convince themselves that it will all be okay.

This, too, makes sense. We're nothing if not adaptable. For myself, if something changes that I didn't want changed, I try to find a way to live with it. Some would call that looking on the positive side of things, I suppose. I call it being too lazy to keep getting fired up about it.

A Harvard University team has done dozens of experiments demonstrating that when we imagine events in the future, we expect the worst of bad events and the best of good events. In reality, bad events don’t make us feel quite so awful and good events don’t make us feel quite so great.

Nah, I also expect the worst of good events. It's a defense mechanism. Then, when things turn out better than I expected, I feel that much happier (or at least less sad) about it.

And while we imagine that if something grave happens to us we won’t be able to cope, or that if something positive happens our lives will be transformed, in both cases we will be the same people we are now. After the initial impact, our emotions will subside and we will feel only a little better or worse than now. The same happens when we’re affected by a policy or situation we don’t like. If it’s possible, we make peace with what we once saw as negative changes.

I'll try to remember that when this November's election doesn't go the way I'd like.

In one sense this is rather hopeful: we try to find the good in every situation. But does this mean that policymakers can do whatever they like and we’ll all decide that’s okay? Not exactly. If that were the case, governments would never get voted out or overthrown in revolutions.

Or maybe I'll remember the "overthrown in revolutions" bit. I don't know.

But when we can’t change everything, making peace with the world might be an important part of our well-being.

Sometimes, though, I think a situation warrants not making peace with the way things are. As I've said before, one way or another: we can't change other people. We can only change ourselves.

The "surprising" bit in the article's headline turns out not to be the case. Dammit, BBC, clickbait? Really? I expected better from you. But I guess I can learn to live with it.

3 Like • Comment • View 1 comment

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